Do some Bone Constraints only work with Root Bones?
The Limit Distance constraint influences the location of the bone. All the bones (except the root bone) are children of another bone in the armature. Because of this you cannot move them (the location is also greyed out in the properties panel). If you want to allow the tip to move, you should un-parent the tip, create an IK constraint on the bone beneath it, and target the tip bone. That should make it work.